81.

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equal to 4 is shown in Figure 4.12 in which, according to Equation (4.39), the membership function centres are 0, 0.33, 0.66 and 1, respectively, λ is computed as 0.33, and a total of sixteen fuzzy subspaces are generated. Note that this classification method generates a unique set of rules for each fuzzy subspace. Each rule in the fuzzy subspace takes the following form:

IF s1 is in A_i and s2 is in A_j

THEN pixel s belongs to class c_ij with strength w_ij

where s1 and s2 are the input features for pixel s, and A_i and A_j (1≤i, j ≤k) are fuzzy subspaces on both dimensions, respectively. The class c_ij and rule strength w_ij based upon the fuzzy rule are determined by the training data. The procedure for generating fuzzy rules in the subspace ij is shown in Figure 4.13a. The first step in Figure 4.13a is to calculate the weighting parameter β_x for each class x using the procedure described in Equation (4.31). We then choose the class c_ij according to its maximal weights among all classes. The rule in fuzzy subspace ij is then generated with strength w_ij calculated in step (3).

For example, in the two-dimensional case, two classes and two fuzzy partitions are generated for each dimension, using Equations (4.38) and

Figure 4.12 An example of fuzzy partitions with k=4 results in total sixteen fuzzy subspaces.

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